Mathematical Modeling, Simulation and Validation of the Dynamic Yarn Path in a Superconducting Magnet Bearing (SMB) Ring Spinning System

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M. Hossain, C. Telke, M. Sparing, A. Abdkader, A. Nocke, R. Unger, G. Fuchs, A. Berger, C. Cherif, M. Beitelschmidt, L. Schultz Mathematical modeling, simulation and validation of the dynamic yarn path in a superconducting magnet bearing (SMB) ring spinning system

Erstveröffentlichung in / First published in: Textile Research Journal. 2017, 87(8), S. 1011 – 1022 [Zugriff am: 07.08.2019]. SAGE journals. ISSN 1746-7748. DOI: https://doi.org/10.1177/0040517516641363

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Review article Textile Research Journal 2017, Vol. 87(8) 1011–1022 ! The Author(s) 2016 Mathematical modeling, simulation and Reprints and permissions: sagepub.co.uk/journalsPermissions.nav validation of the dynamic yarn path in a DOI: 10.1177/0040517516641363 superconducting magnet bearing (SMB) journals.sagepub.com/home/trj ring spinning system

M Hossain1, C Telke2, M Sparing3, A Abdkader1, A Nocke1, R Unger1, G Fuchs3, A Berger3, C Cherif1, M Beitelschmidt2 and L Schultz3

Abstract The new concept of a superconducting magnetic bearing (SMB) system can be implemented as a twisting element instead of the existing one in a ring spinning machine, thus overcoming one of its main frictional limitations. In the SMB, a permanent magnet (PM) ring rotates freely above the superconducting ring due to the levitation forces. The revolution of the PM ring imparts twists similarly to the traveler in the existing twisting system. In this paper, the forces acting on the dynamic yarn path resulting from this new technology are investigated and described with a mathematical model. The equation of yarn movement between the delivery rollers and the PM ring is integrated with the Runge-Kutta method using MATLAB. Thus, the developed model can estimate the yarn tension and balloon form according to different spindle speeds considering the dynamic behavior of the permanent magnet of the SMB system. To validate the model, the important relevant process parameters, such as the yarn tension, are measured at different regions of the yarn path, and the balloon forms are recorded during spinning with the SMB system using a high speed camera.

Keywords mathematical modeling, yarn tension, balloon form, ring spinning, superconducting magnetic bearing

In the existing ring spinning process, the frictional heat ring. The PM ring levitates above the SC ring according generated in the ring/traveler system causes damage to to the principle of levitation. During the spinning pro- both the twisting element and the yarn structure.1 The cess, the yarn (wound onto the bobbin) rotates the PM traveler is not allowed to rotate at more than 50 m/s, ring, instead of the traveler. The patented concept of especially in the case of man-made fibers, due to their the SMB system ensures a smooth running of the spin- melting, caused by the high friction-induced heating, ning process for significantly higher productivity with which limits productivity.2,3 The friction-free supercon- similar yarn properties to the conventional process.5 ducting magnet bearing (SMB) eliminates this restric- tion and thus allows increase of the spindle speed much higher than with existing spinning machines. In our 1 previous work, different concepts of SMB system Institute of Textile Machinery and High Performance Material Technology (ITM), Technische Universita¨t Dresden, Dresden, Germany have been presented, and a suitable one has been suc- 2 4 Chair of Dynamics and Mechanism Design, Institute of Solid Mechanics, cessfully integrated in a ring spinning tester. The SMB Technische Universita¨t Dresden, Dresden, Germany system comprises of two rings, a magnetic element of 3IFW Dresden, Institute for Metallic Materials, Dresden, Germany Neodymium Iron Boron (NdFeB) with a yarn guide attached to it, and a high temperature superconductor Corresponding author: M Hossain, Institute of Textile Machinery and High Performance Material (SC) from YBCO (YBa2Cu3O7-x) chemical compounds. Technology (ITM), Technische Universita¨t Dresden, Hohe Straße 6, The superconductor (SC) ring is cooled down below its Dresden 01069, Germany. critical temperature at a fixed distance from the PM Email: [email protected] 1012 Textile Research Journal 87(8)

However, more investigations are necessary to analyze speed for the conventional ring spinning system, which the physical challenges of the new technology during is significant compared to other existing models.23 the spinning process. The research work is based Moreover, the developed algorithm of sensitivity ana- on the dynamic behavior of the novel SMB system lysis provides a solution space with valid stable initial for the ring spinning process, which is a combination values for the numerical solution, as the equation of of textile and superconducting technology, and motion is highly non-linear. The simulated results, describes the theoretical modeling, simulation, and such as the yarn tension and the balloon forms, have measurement methods for the dynamic yarn path with also been validated with the measured values. In the this magnetic bearing. The dynamic yarn path and the new ring spinning method, the existing ring/traveler resultant forces of the SMB system are theoretically system has been replaced by the SMB system, which modeled with respect to spindle speed, yarn count, bal- changes the yarn tension distribution and balloon loon geometry, and the mass and diameter of the PM geometry during the spinning process. That is why the ring. In the numerical solution, the boundary condi- existing model of ring/traveler system and the dynam- tions of the yarn path with the SMB system are also ical yarn path need to be described afresh, especially for described. This model provides information about the the yarn path in regions III and IV (Figure 1). In this dynamical forces during the spinning process and helps paper, the theoretical model described in Hossain to determine the textile technological and physical limi- et al.23 is modified considering the dynamics of the tation for SMB system. Furthermore, the calculated SMB system in the ring spinning process. yarn tension and corresponding balloon form from For this purpose, at first the dynamics of the rotating this developed model help to evaluate the new system, PM ring during the spinning process have been theor- especially at higher spindle speed. etically modeled. The mathematical modeling of the yarn path between the delivery rollers and the PM Theoretical model ring is similar to that of the conventional one. However, the boundary conditions for the numerical There are many theoretical models for the prediction of solutions derived from the equation of the PM ring the balloon form and its associated yarn tension.6–22 are described in the section below. Crank described the balloon in a quasi-stationary case in cap spinning, considering the centrifugal, Coriolis, 6 Equation of motion of the permanent magnet (PM) and gravitational forces. Mack neglected the Coriolis in the SMB system forces in the equation of motion of yarn, and considered the air drag constant related to the Reynold’s The attainable bearing force, FB, in the SMB system is number.7 He described the equation in non- based on the flux pinning of magnetic fluxes penetrating dimensional form and introduced the inextensibility the superconductor, obtained after field-cooling the condition. However, the boundary condition of the superconducting ring, and can be expressed as24 equation was not clearly shown in his work. De Barr Ã showed how the yarn tension depends on process par- FB B grad B dV ð1Þ ameters, such as the friction between ring and traveler, air drag, balloon form, etc. He also applied the model- This means that the achievable bearing force is pro- ing to the balloon considering the balloon control portional to the maximum flux density, BÃ, and the gra- ring.8–11 Lisini described the balloon dynamics as a dient of the field change in the superconductor. The field two point boundary value problem and presented the distribution of PM rings is arranged in a way that high equation of motion in the ring spinning process in both values of BÃ and gradB can be achieved in the largest the stationary and non-stationary cases.12 Batra solved possible volume of the superconductor (SC). If the SC is the non-dimensional equations by integrating the deflected in the SMB system, an attractive or repulsive dynamics of traveler using the Runge-Kutta force acts to hold the PM in its original position.25 method.13–16 Fraser explored the relationship between The SMB system is mathematically described as a guide eye tension and traveler mass, mainly in the case spring–mass–damper system, in which the PM ring is of free balloon spinning.17 He solved the equations driven by the spindle via the yarn attached to it. The numerically in a dimensionless form to avoid the com- yarn tension acting at the yarn guide of the PM ring can plexity of several influencing parameters, such as the be described with the help of Figure 1(b). T1 and T2 are spindle speed. Recently, we have described the equation the yarn tensions at the yarn guide of the PM ring on of motion in dimensioned form, considering the the balloon side and on the cop side, respectively. The Coriolis force, and have solved the equation numeric- total yarn tension, TG, that is, the sum of the yarn ten- ally to discover the actual yarn tension and balloon sions T1 and T2, results in the radial, axial, and tilt form in non-normalized values with respect to spindle displacement of the PM ring. This displacement is Hossain et al. 1013

(a) Drafting system (b)

Delivery rollers Yarn guide

II Yarn guide at PM

Cop with Yarn III Permanent magnet ring Superconductor ring Ring rail IV

Spindle

Figure 1. The ring spinning process with a superconducting magnetic bearing (SMB). (a) Schematic diagram of spinning and definition of yarn path at different regions23: I—between delivery rollers and yarn guide; II—between yarn guide and yarn guide at PM; III—through yarn guide of PM; IV—between yarn guide of PM and cop. (b) Definition of coordinate systems and yarn path. counteracted by the strong restoring forces of the SMB is determined by equilibrium between the yarn force, z r system—FB in the axial, and FB in the radial directions. TG, and the damping. This damping of the PM ring There exists no restoring force in the circumferential influences the motion of the ring, and is responsible direction for this round SMB geometry, thus the total for the necessary slip between spindle and ring. external yarn tension, TG, in the circumferential direc- In the case of free rotation without external yarn tion acts as a driving force for the free rotation of the force—that is, when TG ¼ 0 (e.g. during end-breakages) PM ring. The amplitudes of displacement for the dif- equation (2) simplifies to ferent directions of yarn forces are defined by the bear- 25 ing stiffness. !_ PM þ 2rot!PM ¼ 0 ð4Þ The rotation of the PM ring is thus described in equation (2), according to the principle of angular with momentum d ¼ R ð5Þ _ rot !PMJR þ dR!PM ¼ TGðÞt aiPM ð2Þ 2JR where !PM is the angular velocity of the PM ring, where rot is the decay constant of the damped oscilla- !_ PM d ð!PMÞ=dt, JR is the moment of inertia, dR is the damping constant of rotation of the rotating PM tion of the PM ring without external forces. The yarn- driven rotational motion of the PM-ring is decelerated ring, and aiPM is the inner radius of the PM ring. Air friction and dissipative processes in the superconductor by air friction and the viscous motion of the flux lines damp the rotation of the PM ring. For the quasi- (vortices) from the superconductor. This is described by stationary case, the angular acceleration of the PM the rotational decay constant, rot, in equation (5). This constant is determined by accelerating the PM in the ring is !_ PM ¼ 0 and, therefore, equation (2) can be writ- ten as ring spinning machine up to the defined spindle speeds, such as 5000, 20,000, etc. The decelerations of the PM ring are measured over time from the defined spindle ! dR PM ¼ TGaiPM ð3Þ speed to still stand after cutting the yarn. The constant angular frequency of the PM ring Mathematical formulation of the yarn path TGðtÞaiPM In the mathematical description of the yarn path, the !PM ¼ dR yarn is considered as a one dimensional continuum with 1014 Textile Research Journal 87(8) homogeneous and circular cross section, so that only where v is the delivery velocity of the yarn at the deliv- yarn tension acts in the yarn. The other important ery rollers. assumptions and limitations of the presented model Equation (7) is a second order differential equation are as follows. system for the yarn path at point L. The boundary conditions for this mathematical problem at the – The yarn is considered to be inextensible. yarn guide of the PM ring, considering the dynamics of the PM ring, have to be defined for the numerical – The influences of centrifugal force and Coriolis force solution. are considered. – The dynamic model during synchronous running of the PM is considered. Boundary conditions for the numerical solution of – The friction between yarn and the yarn guide at the SMB system PM is considered constant. The first boundary condition at the yarn guide is as – The movement of the ring rail is not considered. follows. – Balloon control ring is not considered. For the guide eye (O (Figure 1(b)), the boundary conditions are23 – The effect of yarn twist is neglected. rðÞ¼0 10À3m, ðÞ¼0 0, zðÞ¼0 0 ð8Þ As shown in Figure 1(a), the yarn path from the delivery roller to the winding point of the cop has The first boundary condition at the yarn guide of the been divided in four regions. The theoretical modeling PM ring, N, (Figure 1(b)) is of the dynamic yarn path in regions I and II is partly based on the existing mathematical models of the con- rsðÞ¼l aiPM, zsðÞ¼l h ð9Þ ventional ring spinning process,23 which is developed taking into account the above-mentioned assumptions where sl describes the length of the yarn between yarn and conditions. The dynamic forces, such as centrifugal guide and the yarn guide of PM ring at N and h is the forces, air resistance forces, and the Coriolis force, balloon height (Figure 1(b)). mainly govern the geometry of the dynamic yarn path The second boundary condition at the yarn guide of and control the tension distribution along the yarn the PM ring can be defined from the equation of path. In order to define the yarn path, Figure 1(b) motion of the PM ring (equation (3)), where the left describes a rotating cylindrical coordinate system of side of the equation describes the components of the rsðÞ, ðÞs , zsðÞ corresponding to the unit vectors yarn tension, T1 and T2, in the circumferential direction er, eh, ez, which rotates with the angular speed of the (Figure 1(b)) PM ring and describes the dynamic forces. If a material R e e 0 point, L, is defined as position vector ðÞ¼s, t r r þ z z, T1½g sin À aiPM ðsl Þ aiPM ¼ dR!PM ð10Þ the equation of motion of yarn dynamics can be for- mulated as follows, according to the well-known equa- where T1 is the yarn tension at the yarn guide of the PM 17 tion of motion ring on the balloon side and gð¼ ey ¼1.7) is the frictional parameter between yarn and yarn guide of the D2 D 2 PM ring as constant, ðÞ0 d ðÞ=ds.17 The yarn tension at m R þ 2!PMez Â R þ !PMez Â ðÞez Â R the yarn guide of the PM ring on the cop side, T ÈÉ@ @ 2 ¼ T R þ F ð6Þ (Figure 1(b)), is estimated by the Euler equation, as @s @s follows

where m is the yarn mass in g/km, !PM is the angular y T2 ¼ T1 Á e ð11Þ velocity of the PM ring, D is the differential operator, and F denotes the air drag force per unit length. If the problem is considered as quasi-stationary, equation (6) where y is the frictional coefficient between the yarn can be expressed as follows, in dimensioned form and the yarn guide of the PM ring and is the angle shown in Figure 1(b). The frictional parameter is con- 2 sidered constant for this modeling, as mentioned in the 2 d R dR 2 mv þ 2! vez Â þ ! ez Â ðÞez Â R assumptions. However, the new concept needs to be ds2 PM ds PM developed to measure the friction between the yarn 2 dR d R and the yarn guide of the PM ring during the spinning ¼ T 0 þ T þ F ð7Þ ds ds2 process at different spindle speeds. Hossain et al. 1015

Numerical solution of the theoretical respect to spindle speed. The balloon forms also model for SMB system increase due to the centrifugal forces with respect to spindle speeds from 5000 to the 20,000 r/min, which is The equation of motion of the yarn path described in shown in Figure 3(a). The quantitative values of max- equation (7) is integrated with the Runge-Kutta imum balloon diameter according to spindle speed are method with a spatial step size of 0.0001 using also shown in Appendix 2. As the implementation of MATLAB, and optimized using a shooting method so the SMB system is aimed at spinning yarn at higher that the above-mentioned boundary conditions (in spindle speed than the existing ring traveler system, equations (8–10) are fulﬁlled.27 Finally, an iterative the aim of the modeling is to gather information optimization method with the Levenberg–Marquardt about the changes of yarn tension and balloon forms algorithm is applied to minimize the squared error resi- with the new SMB system, especially considering the dual sum of the reformulated boundary conditions. The spindle speeds. The results can be considered for the boundary conditions described in equations (9) and stationary case when the yarn PM ring and spindle (10) are reformulated to obtain the residual À1 and run synchronously. À2, with the weighted function, k, as follows Characterization of the developed model À 0 2 1 ¼ 1ðÞrsðÞl, T0, r ÀaiPM ð12Þ for SMB system

y 0 0 2 In order to characterize the dynamic yarn path and to À2 ¼ 2½T1½e sin À aiPM ðÞsl, T0, r aiPM À dR!PM validate the presented model, the yarn tensions are ð13Þ measured at different regions, such as in regions I, II, and IV (Figure 1(a)). where T0 is the yarn tension at the guide eye (O,as The goal is to measure the important physical fac- shown in Figure 1(b)), T1 is the yarn tension at the tors influencing the yarn and SMB system, and to yarn guide of the PM ring on the balloon side, !PM is explore their interaction with respect to different spin- the velocity of the PM ring, and the angle shown in dle speeds. The different principles of yarn tension Figure 1(b). measurement in region I, II, and IV have already The sum of residual results is been implemented in the conventional ring spinning process with the ring/traveler system.28 After some N modifications, the sensors are further applied in the À 0 À 0 ðÞsl, T0, r ¼i iðÞsl, T0, r ð14Þ case of SMB spinning. Moreover, the balloon forms Xi¼1 are recorded with the high speed camera at different spindle speeds. During the optimization, the start values, T0ð0Þ and Before the measurement of the yarn tension, the ring r0ð0Þ, are varied so that minimum errors can be achieved spinning tester available at the Institute of Textile in relation to the boundary conditions Machinery and High Performance Material Technology (ITM), Dresden was constructively modi- N fied for the special technological requirements. The ring min ÀðÞs ,T ,r0 ¼min À ðÞðs ,T ,r0 15Þ 0 l 0 0 i i l 0 rail, spindle, front casing behind the spindle, etc., of the T0,r T0,r Xi¼1 existing ring spinning tester were replaced with non- magnetic materials, so that the magnetic forces of the The parameters considered for the numerical solu- PM ring and the levitation forces of the SMB system tion and for the experiments are listed in Appendix 1. were not magnetically influenced. The cryostat used for From this numerical solution, the yarn tensions are the cooling and thermal insulation of the SMB system solved for regions I, II, and III. The yarn tension at was integrated on the ring rail of the spinning tester, as region IV is calculated, considering a constant frictional shown in Figure 2. The superconducting ring mounted parameter between the yarn and the yarn guide of the in the cylindrical casing must remain cold below its PM ring, using equation (11). The results from the transitions temperature (À196C). For this purpose, numerical solution of the presented model predict the connection system of the casing ensures a continu- the yarn tension and balloon form with different spin- ous supply of liquid nitrogen (LN2) from a cryogenic dle speeds from 5000 to 25,000 r/min (Appendix 2 and dewar with a definite pressure and flow rate of LN2. Figure 3(a)). In Appendix 2, the yarn tension for The temperature inside the cryogenic system is continu- regions I–IV and the geometry corresponding to the ously monitored by a thermo-element. An insulating balloon are presented according to the different spindle process is maintained between the superconductor speeds. The yarn tensions at all regions increase with and the outer cryostat surface by constant pumping 1016 Textile Research Journal 87(8)

distance in the pixels ﬁeld will be then automatically

Cop be ﬁlled in, based on the length of the line selection. The actual diameter of the balloon can be measured Connection of vacuum pump similarly (Figure 4). The balloon forms are also listed PM ring for spindle speeds from 5000 to 20,000 r/min in Table 1.

N2 Gas discharge According to Figure 3(b) and Table 1, the balloon Cylindrical casing forms increase with respect to spindle speed. Thermo-element

LN2 supply Yarn tension between delivery rollers and yarn guide (Region I) Figure 2. Integration of superconducting magnetic bearing A three-point sensor from Tensometric Messtechnik (SMB) and cryostat in a ring spinning tester. GmbH is used for the measurement of tension in region I at different spindle speeds (Figure 5(a)). The yarn path acting radially on the measuring roller with a vacuum pump. To fix the initial levitation dis- of the sensor deforms the complex-shaped cantilever tance the PM ring is placed above the superconductor attached to it. The full bridge of built-in strain gauges (SC), positioned with three distance pieces on the top of converts this deformation into a proportional electrical the cryogenic system. The PM ring is then centered with output signal, which is recorded using LabView Signal the help of a centering device with respect to the spin- Express. During calibration, dead weights of 10 g and dle. When the temperature falls to À196C, the distance 20 g are hung over the spun yarn to set the scale in the pieces are removed. The PM ring is now free to rotate data acquisition program. The experiment is repeated over the SC, or rather on the cylinder casing, and is in the SMB ring spinning tester to verify the measured ready to start the spinning process with the SMB values. The mean values of yarn tension for different system. spindle speeds are shown in Figure 5(b). According to Figure 5(b), yarn tension increases with higher spindle Balloon form (Region II) speed in both in the calculated and experimental results. The measured values are close to the calculated ones, The recording of balloon forms is carried out by using a especially at the spindle speeds of 5000 and high speed camera (Photron Fastcam Ultima SA-3, 10,000 r/min. As the spindle speed increases further, frame rate 12.5 KHz for 1024 Â 1024 pixel) at the such as at 15,000 r/min, the vibration of the PM ring ITM-ring spinning tester to determine the influence of causes a yarn tension increment and generates peaks, centrifugal force and air resistance on the balloon form. which is still more prominent at 20,000 r/min. A black reference grid is fixed behind the spindle to Moreover, the twist propagation of the yarn from the avoid the reflection of light during the experiment. PM ring to the delivery rollers is retarded by the mea- The balloon forms are measured at spindle speeds of suring sensor, and reduces the strength of the yarn, 5000, 10,000, 15,000, and 20,000 r/min, and at different resulting in frequent end-breakages. For this reason, ring rail positions, as shown in Figure 3(b). According measurement has not been carried out for higher to the Figure 3(b), the position of maximum radius of speeds, such as 25,000 r/min. Modification of the balloon is located in the plane of PM ring at lower sensor with the rotating guiding elements and measur- spindle speed such as at 5000 r/min. At higher spindle ing roller should solve the problem of continuous end- speed, the maximum radius of balloon locates in the breakages during measurement of yarn tension. middle of the yarn guide and PM ring due to higher centrifugal force. Yarn tension between yarn guide and PM ring The recorded images of balloon form have been ana- (Region II) lyzed to measure the maximum balloon diameter using the software Image J.29 Each maximum balloon form In order to measure the yarn tension in the bal- for a defined spindle speed has been measured several loon—that is, between the yarn guide and the yarn times with this program. For the measurement of bal- guide of the PM ring—a sensor from Tensometric loon form with Image J, at first a straight line is drawn GmbH, Germany, has been modified. The sensor con- on the bobbin diameter of the image. The program rec- sists of a measuring roller and two guiding rollers, ords the x, y-coordinates and the length of the line in which are fixed to the sensor part. The measuring pixels, to define the spatial length of the active image of roller measures the radial force of the rotating balloon the balloon form in calibrated units. The scale can be acting on it, and the two guide rollers keep the yarn set to enter the known diameter of the cop. The path deflection constant. Hossain et al. 1017

Figure 3. (a) Balloon forms from numerical results. (b) The recorded balloon forms at different spindle speeds.

During calibration of the sensor, different dead Yarn tension between the yarn guide of the PM ring weights, such as 10 g and 20 g, are hung on the spun and the winding point of cop (Region IV) yarn, so that the yarn presses against the measuring roller. The corresponding yarn tension in the longitu- As shown in Figure 7(a), the measuring ring (7), which dinal direction can be measured through the radial is mounted concentrically between the PM ring (5) and force. The sensor is positioned in such a way that the the cop (8), is constructed from non-magnetic material balloon touches the measuring roller and the guiding to avoid magnetic influences during measurement element (Figure 6(a)). The results are recorded in a data (Figure 7(a)). The yarn path from the yarn guide of acquisition LabView program and are analyzed with the PM to the winding point of the cop presses against DIAdem. According to Figure 6(b), the mean value the measuring ring and deflects the spring leaves of yarn tension increases according to the spindle mounted under the measuring ring. Moreover, four speed, as expected. The mean yarn tension at the spin- spring leaves with strain gauges are mounted under dle speeds of 5000 to 20,000 r/min are found to be from the measuring ring. The deflection of the strain 5 cN to 55 cN, with 2% variation, respectively (Figure gauges attached to these leaves changes the electrical 6(b)). However, the balloon deforms to a certain level signal, which is recorded with a data acquisition pro- after touching the measuring roller of the sensor using gram LABView Signal Express. this method, especially at higher spindle speed, which During the calibration, the dead weight, of 10 and influences the measured values. 20 g are hung over the measuring ring with spun yarn, 1018 Textile Research Journal 87(8)

and thus set the scale in LabView SignalExpress, accordingly. The mean values of measured yarn tension are presented in Figure 7(b) for diﬀerent spindle speeds. According to Figure 7(b), the yarn tension increases with respect to spindle speed from 5000 to 20,000 r/min. However, the variation of the yarn tension during the measurements can be attributed to the dynamic friction between the rotating yarn and the measuring ring, and also due to the eccentricity of the measuring ring. In Table 2, the mean measured yarn tensions at all yarn regions have been summarized. The highest yarn tension occurs in region IV between the yarn guide of PM ring and cop. Moreover, the variation of yarn tension increases especially at higher spindle speed, due to the vibration of yarn and PM ring.

Validation of yarn tension and maximum balloon form between measured and calculated values In this section, the calculated yarn tensions are com- Figure 4. Measurement of maximum balloon diameter at the pared with measured ones for the regions I, II, and IV spindle speed of 15,000 r/min. for spindle speed, of 5000–20,000 r/min. The measured and calculated maximum balloon diameters are also compared, according to spindle speed. Table 1. Measured values of maximum balloon diameter The difference—that is, the absolute error between the calculated and measured yarn tension—is presented Spindle speed Mean value of Standard in Figure 8 and in Appendix 3 for the spindle speeds of (r/min) diameter (mm) deviation 5000–20,000 r/min. 5000 45.7 0.2 The calculated and measured balloon forms are also 10,000 57.5 0.3 illustrated in Figure 3, and the maximum balloon diameters are quantitatively compared in Table 3 for differ- 15,000 60.8 0.3 ent spindle speeds. According to Figure 8 and Table 3, 20,000 68.7 0.4 the difference between the calculated and measured

Figure 5. (a) Measurement set-up for yarn tension in region I between delivery rollers and yarn guide. (b) Mean values of yarn tension in region I at different spindle speeds. Hossain et al. 1019 yarn tension increases with spindle speeds higher than 15,000 r/min (Figure 8, Table 3). The errors can be Table 2. Experimental mean values of yarn tension at different attributed to the simplification of the assumptions in regions with respect to spindle speeds the model, such as considering the material behavior Mean value of measured yarn tension of the yarn as linear. Moreover, the induced vibrations Spindle of the PM and its influence on yarn are not considered speed, F(I), Std. F(II), Std. F(IV), Std. in the model presented. r/min cN dev cN dev cN dev 5000 7.0 0.9 6.8 0.1 11.8 1.4 Conclusions and outlook 10000 17.5 1.5 15.4 0.2 29.5 1.8 A theoretical model of the dynamic yarn path in the 15000 35.4 1.8 30.0 0.3 58.5 2.8 SMB ring spinning process was developed for the 20000 57.0 2.9 57.0 0.5 95.0 3.3 quasi-stationary case. The presented model was further F(I): yarn tension between delivery rollers and yarn guide in region I; F(II): modified based on the previous model of a ring traveler yarn tension between yarn guide and PM ring in region II; F(IV): yarn system. For this purpose, the equation of motion of the tension between PM ring and cop in region IV.

Figure 6. (a) Sketch28 and experimental set-up of modified sensor used for the measurement of yarn tension in region II between the yarn guide and the yarn guide of the PM ring. (b) Mean values of yarn tension in region II at different spindle speeds. F (II): Yarn tension in region II; FN: Radial component of the yarn tension.

Figure 7. (a) Set-up for the measurement of yarn tension in region IV between the yarn guide of the PM ring and the cop. (b) Mean values of yarn tension in region IV at different spindle speeds. 1: ring rail; 2, 3: support for measuring ring; 4: cryostat; 5: permanent magnet ring; 6: superconductor; 7: measuring ring with spring leaves and strain gauges; 8: cop. 1020 Textile Research Journal 87(8)

Figure 8. Comparison of yarn tension between measured and calculated values for (a) region I; (b) region II; (c) region IV.

Table 3. Comparison of maximum balloon diameter between tension and the balloon forms, showed a comparable calculated and measured values correlation with the calculated values from the developed model. However, the experimental yarn tension Spindle Maximum calculated Maximum measured Absolute speed balloon diameter balloon diameter error was generally higher than the calculated value, espe- (r/min) (Appendix 2) (mm) (Table 1) (mm) (mm) cially at higher spindle speeds, because some assumptions, such as the vibration of PM ring during the 5000 45.4 45.7 0.3 spinning process and the resulting vibration of the 10,000 54.0 54.5 0.5 yarn, were not considered in the mathematical model. 15,000 59.8 60.8 1.0 The difference between the calculated and measured 20,000 67.4 68.7 1.3 yarn tension at all regions increased due to the vibration of the PM, especially at higher spindle speed. Further, the friction between the yarn and the yarn rotating permanent magnet (PM) was at first described guide of the PM were considered constant in the as a spring–mass–damper system. The necessary speed numerical solution, but they varied during the experi- difference between the spindle and PM ring was derived ments. In further work, the vibration of the yarn and with the damping parameter—that is, with the rota- the PM ring needs to be considered in the numerical tional decay constant for different spindle speeds. The model, especially for higher spindle speeds. Moreover, equation of motion of the dynamic yarn path was pre- the process parameters between the conventional and sented in non-normalized values (with physical units). the SMB spinning system, for both the calculated and For numerical solution of the model, the equations measured values, should be compared. were solved with the Runge-Kutta method, using the MATLAB program. The boundary conditions at the Declaration of conflicting interests yarn guide of the PM ring were modified considering The authors declared no potential conflicts of interest with the dynamic behavior of the SMB system. The numer- respect to the research, authorship, and/or publication of this ical results show the calculated yarn tension at different article. regions of the yarn path, and the corresponding balloon forms, according to spindle speed from 5000 to Funding 20,000 r/min. The yarn tension and balloon form gen- The authors disclosed receipt of the following financial sup- erally increase according to spindle speed, similar to the port for the research, authorship, and/or publication of this conventional spinning process. article: This research is funded by German Research In order to validate the developed model, the yarn Foundation, DFG (Project No. CH 174/33-1 and SCHU tensions at different regions were also measured with 1118/12-1). The authors would like to thank DFG for the modified sensors in consideration of the SMB spinning financial support. system, and the values were analyzed at different spindle speeds. A high speed camera was used to record the References balloon forms with corresponding spindle speeds. The 1. Wulfhorst B, Gries T and Veit D. Textile technology an diameter of the recorded balloon form for each spindle introduction. Munich: Carl Hanser Verlag, 2015. speed was quantitatively analyzed with Image J soft- 2. Klein W. The technology of short-staple pinning: Short- ware. Statistical analysis, such as for absolute error staple spinning series – Volume 1. Manchester: The and standard deviation, for both the measured yarn Textile Institute, 1998. Hossain et al. 1021

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Appendix 2 Numerical results about yarn tension and balloon form

F(II) Balloon Parameter in region II

Exp. Spindle speed F(I) (cN) (mm) F(III) F(IV)

() (r/min) (cN) h ¼ 180 (mm) hmax (mm) rmax (mm) (cN) (cN) 1 5000 6.5 6.3 163.9 22.7 6.4 11.0 2 10,000 16.6 14.8 129.6 27.0 16.0 28.1 3 15,000 33.5 28.5 121.7 29.9 32.2 56.9 4 20,000 54.6 54.5 114.9 33.7 52.4 92.8 5 25,000 78.7 60.2 106.5 41.2 75.2 133.8

F(I): yarn tension between the clamping point of delivery rollers and yarn guide; F(II): yarn tension between guide eye and yarn guide at PM; F(III): yarn tension at yarn guide of PM; F(IV): yarn tension between yarn guide of PM and winding point of cop; h: balloon height from yarn guide to the yarn guide of PM; rmax: maximum balloon diameter; hmax: balloon height from yarn guide to PM at rmax.

Appendix 3 Comparison of calculated values and measured mean values with absolute errors

F(I) (cN) F(II) (cN) F(IV) (cN) Spindle speed Measured Calculated Absolute Measured Calulated Absolute Measured Calculated Absolute (r/min) values values error values values error values values error

5000 7.0 6.5 0.5 6.8 6.3 0.5 11.8 11 0.8 10,000 17.5 16.6 0.9 15.4 14.8 0.6 29.5 28.1 1.4 15,000 35.4 33.5 1.9 30.0 28.5 1.5 58.5 56.9 1.6 20,000 57.0 54.6 2.4 57.0 54.5 2.5 95 92.8 2.2